Введение в теорию вероятностей. Учебное пособие
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ɉɟɪɜɨɦɭ ɭɪɚɜɧɟɧɢɸ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɫɥɟɞɭɸɳɢɟ ɩɚɪɵ ɧɚɬɭɪɚɥɶɧɵɯ ɱɢɫɟɥ ɛɨɥɶɲɢɯ ɟɞɢɧɢɰɵ Ɉɞɧɚɤɨ ɧɢ ɨɞɧɚ ɢɡ ɧɢɯ ɧɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɜɬɨ ɪɨɦɭ ɭɪɚɜɧɟɧɢɸ ɦɨɠɧɨ ɬɚɤɠɟ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɪɟɲɢɬɶ ɫɢɫɬɟɦɭ ɱɬɨɛɵ ɭɛɟ ɞɢɬɶɫɹ ɱɬɨ ɭ ɧɟɺ ɧɚɬɭɪɚɥɶɧɵɯ ɪɟɲɟɧɢɣ ɧɟɬ
Ɉɬɜɟɬ ɧɟɜɨɡɦɨɠɧɨ
10. Повторение опытов. Формула Бернулли
ɉɭɫɬɶ ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɨɩɵɬ ɩɨɜɬɨɪɹɟɬɫɹ n ɪɚɡ ɢ ɩɪɢ ɷɬɨɦ ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ Ⱥ ɜ ɤɚɠɞɨɦ ɬɚɤɨɦ ɨɩɵɬɟ ɨɫɬɚɟɬɫɹ ɧɟɢɡɦɟɧɧɨɣ ɪɚɜɧɨɣ ɧɟ ɤɨɬɨɪɨɦɭ ɪ ɩɭɫɬɶ q ɪ Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɦɨɞɟɥɶ ɢɫɩɨɥɶɡɭɟɦɚɹ ɞɥɹ ɨɩɢɫɚ
ɧɢɹ ɬɚɤɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɨɩɵɬɨɜ ɧɚɡɵɜɚɟɬɫɹ ɫɯɟɦɨɣ Ȼɟɪɧɭɥɥɢ Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ Pn k ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɫɨɛɵɬɢɟ Ⱥ ɩɨɹɜɢɬɫɹ ɪɨɜɧɨ N
ɪɚɡ ɜ Q ɨɩɵɬɚɯ ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɝɨɜɨɪɹɬ ɬɚɤɠɟ ɨ k ©ɭɫɩɟɯɚɯª ɜ n ɨɩɵɬɚɯ ɂɦɟɟɬ ɦɟɫɬɨ ɫɥɟɞɭɸɳɚɹ ɮɨɪɦɭɥɚ Ȼɟɪɧɭɥɥɢ
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Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ Ɏɢɤɫɢɪɭɟɦ ɤɚɤɢɟ ɥɢɛɨ k ɨɩɵɬɨɜ Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ An k ɫɨɛɵɬɢɟ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ A ɪɨɜɧɨ k ɪɚɡ ɬ ɟ ɜ ɷɬɢɯ ɜ k ɨɩɵɬɚɯ ɩɪɨɢɫ ɯɨɞɹɳɟɟ ɫɨɜɦɟɫɬɧɨ ɫ ɧɟɧɚɫɬɭɩɥɟɧɢɟɦ A ɧɚɫɬɭɩɥɟɧɢɟɦ A n k ɪɚɡ Ɉɱɟɜɢɞ ɧɨ ɱɬɨ
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ɋɨɛɵɬɢɟ Bn k ɫɨɫɬɨɹɳɟɟ ɜ ɬɨɦ ɱɬɨ Ⱥ ɩɨɹɜɢɬɫɹ ɪɨɜɧɨ k ɪɚɡ ɪɚɜɧɨɫɢɥɶɧɨ ɧɚɫɬɭɩɥɟɧɢɸ ɤɚɤɨɝɨ ɥɢɛɨ ɢɡ ɨɩɢɫɚɧɧɵɯ ɩɨɩɚɪɧɨ ɧɟɫɨɜɦɟɫɬɧɵɯ An k ɬ ɟ ɢɯ
ɫɭɦɦɟ Ɍɨɝɞɚ |
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ȼ ɭɫɥɨɜɢɹɯ ɩɭɧɤɬɚ ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ A ɜ Q ɨɩɵɬɚɯ ɨɬ k ɞɨ k ɪɚɡ ɬ ɟ ɢɥɢ k ɢɥɢk ɢɥɢ k ɟɫɬɶ ɨɱɟɜɢɞɧɨ
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ɑɢɫɥɨ ɬo ɩɨɹɜɥɟɧɢɹ ɫɨɛɵɬɢɹ ɧɚɡɵɜɚɟɬɫɹ ɧɚɢɜɟɪɨɹɬɧɟɣɲɢɦ ɟɫɥɢ ɜɟ ɪɨɹɬɧɨɫɬɶ ɩɨɹɜɥɟɧɢɹ ɫɨɛɵɬɢɹ ɬo ɪɚɡ ɜ Q ɢɫɩɵɬɚɧɢɹɯ ɩɪɟɜɵɲɚɟɬ ɢɥɢ ɩɨ ɤɪɚɣ ɧɟɣ ɦɟɪɟ ɧɟ ɦɟɧɶɲɟ ɜɟɪɨɹɬɧɨɫɬɢ ɨɫɬɚɥɶɧɵɯ ɜɨɡɦɨɠɧɵɯ ɢɫɯɨɞɨɜ ɢɫɩɵɬɚɧɢɣ Ɇɨɠɧɨ ɞɨɤɚɡɚɬɶ ɱɬɨ ɱɢɫɥɨ ɬo ɢɥɢ ɧɚɢɜɟɪɨɹɬɧɟɣɲɢɟ ɱɢɫɥɚ ɩɨɫɤɨɥɶɤɭ ɢɯ ɦɨ ɠɟɬ ɛɵɬɶ ɢ ɞɜɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɞɜɨɣɧɨɝɨ ɧɟɪɚɜɟɧɫɬɜɚ
n p d ɬo d Q S
Ɉɛɨɛɳɟɧɢɟɦ ɫɯɟɦɵ Ȼɟɪɧɭɥɥɢ ɹɜɥɹɟɬɫɹ ɫɥɟɞɭɸɳɚɹ ɫɯɟɦɚ ɧɚɡɵɜɚɟɦɚɹ ɩɨɥɢɧɨɦɢɚɥɶɧɨɣ ɉɭɫɬɶ ɨɬ ɨɩɵɬɚ ɤ ɨɩɵɬɭ ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵ ɬɢɹ Ⱥ ɦɨɠɟɬ ɦɟɧɹɬɶɫɹ ɛɭɞɭɱɢ ɡɚɜɢɫɢɦɨɣ ɬɨɥɶɤɨ ɨɬ ɧɨɦɟɪɚ ɨɩɵɬɚ ɬɚɤ ɱɬɨ ɜ ɩɟɪɜɨɦ ɨɩɵɬɟ ɨɧɚ ɪɚɜɧɚ p ɜɨ ɜɬɨɪɨɦ ± p ɜ n ɦ ± pn ɩɨɥɨɠɢɦ q p q p qn pn Ɍɨɝɞɚ ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ Ⱥ
ɜ ɷɬɢɯ n ɨɩɵɬɚɯ ɪɨɜɧɨ k ɪɚɡ ɨɤɚɡɵɜɚɟɬɫɹ ɪɚɜɧɨɣ ɤɨɷɮɮɢɰɢɟɧɬɭ ɩɪɢ zk ɜ ɦɧɨɝɨ ɱɥɟɧɟ
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ɮɭɧɤɰɢɸ Mn z ɧɚɡɵɜɚɸɬ ɩɪɨɢɡɜɨɞɹɳɟɣ
ȿɫɥɢ ɠɟ ɜɫɟ p j p ɨɞɢɧɚɤɨɜɵ ɢ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɨɞɢɧɚɤɨɜɵ ɜɫɟ q jp j q ɬɨ ɫɨɝɥɚɫɧɨ ɮɨɪɦɭɥɟ ɛɢɧɨɦɚ ɇɶɸɬɨɧɚ ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɢ zk ɫɨɜ
ɩɚɞɺɬ ɫ ɩɪɚɜɨɣ ɱɚɫɬɶɸ ɜ ɪɟɡɭɥɶɬɚɬɟ ɱɟɝɨ ɩɪɢɯɨɞɢɦ ɤ ɮɨɪɦɭɥɟ Ȼɟɪɧɭɥɥɢ
11. Предельные теоремы в схеме Бернулли
ɋ ɪɨɫɬɨɦ n n of ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɮɨɪɦɭɥɵ Ȼɟɪɧɭɥɥɢ ɢ ɟɟ ɫɥɟɞɫɬɜɢɣ ɩɪɢɜɨɞɢɬ ɤ ɝɪɨɦɨɡɞɤɢɦ ɜɵɱɢɫɥɟɧɢɹɦ Ɋɚɫɫɦɨɬɪɢɦ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɨɨɬɜɟɬɫɬɜɭ ɸɳɢɟ ɩɪɢɛɥɢɠɟɧɧɵɟ ɮɨɪɦɭɥɵ
ɉɪɢ ɛɨɥɶɲɢɯ ɡɧɚɱɟɧɢɹɯ n ɢ p ɡɧɚɱɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɢ Ȼɟɪɧɭɥɥɢ Ɋn k ɦɨɠɧɨ ɩɪɢɛɥɢɠɟɧɧɨ ɜɵɱɢɫɥɢɬɶ ɩɨ «ɥɨɤɚɥɶɧɨɣ» ɮɨɪɦɭɥɟ Ʌɚɩɥɚɫɚ
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Ʌɨɤɚɥɶɧɚɹ ɮɨɪɦɭɥɚ Ʌɚɩɥɚɫɚ ɩɪɢ n of |
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d const ɜɵɬɟɤɚɟɬ ɢɡ ɩɪɟɞ |
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ɫɬɚɜɥɟɧɢɹ ɚɫɢɦɩɬɨɬɢɱɟɫɤɨɣ ɮɨɪɦɭɥɵ |
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ȿɫɥɢ ɤɨɥɢɱɟɫɬɜɨ n ɨɩɵɬɨɜ ɜɟɥɢɤɨ ɢ p ɬɨ ɜɟɪɨɹɬɧɨɫɬɶ
Pn k d k d k ɬɨɝɨ ɱɬɨ ɫɨɛɵɬɢɟ Ⱥ ɩɪɨɢɡɨɣɞɟɬ ɧɟ ɦɟɧɟɟ N ɢ ɧɟ ɛɨɥɟɟ N ɪɚɡ ɦɨɠɟɬ ɛɵɬɶ ɧɚɣɞɟɧɚ ɩɨ ɩɪɢɛɥɢɠɟɧɧɨɣ ɢɧɬɟɝɪɚɥɶɧɨɣ ɮɨɪɦɭɥɟ Ʌɚɩɥɚɫɚ
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Ɍɚɛɥɢɰɵ ɡɧɚɱɟɧɢɣ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɣ ɮɭɧɤɰɢɢ M ɯ ɢ ɢɧɬɟɝɪɚɥɶɧɨɣ ɮɭɧɤ ɰɢɢ Ɏ ɯ ɢɦɟɸɬɫɹ ɜɨ ɦɧɨɝɢɯ ɥɢɬɟɪɚɬɭɪɧɵɯ ɢɫɬɨɱɧɢɤɚɯ ɫɦ ɧɚɩɪ > @ ɉɪɢ ɷɬɨɦ ɢɫɩɨɥɶɡɭɟɬɫɹ ɫɜɨɣɫɬɜɨ ɱɟɬɧɨɫɬɢ ɮɭɧɤɰɢɢ M M ɯ M ɯ ɢ ɫɜɨɣɫɬɜɨ ɧɟ ɱɟɬɧɨɫɬɢ Ɏ ɯ Ɏ ɯ ± Ɏ ɯ
ɂɧɬɟɝɪɚɥɶɧɚɹ ɮɨɪɦɭɥɚ Ʌɚɩɥɚɫɚ ɜɵɬɟɤɚɟɬ ɢɡ ɚɫɢɦɩɬɨɬɢɱɟɫɤɨɝɨ ɩɪɟɞɫɬɚɜ ɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ Ȼɟɪɧɭɥɥɢ ɢ ɫɥɟɞɫɬɜɢɹ ɮɨɪɦɭɥɵ Ȼɟɪɧɭɥɥɢ
ɋ ɩɨɦɨɳɶɸ ɢɧɬɟɝɪɚɥɶɧɨɣ ɮɨɪɦɭɥɵ Ʌɚɩɥɚɫɚ ɦɨɠɧɨ ɧɚɣɬɢ ɜɟɪɨɹɬ ɧɨɫɬɶ ɨɬɤɥɨɧɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɱɚɫɬɨɬɵ w A ɫɨɛɵɬɢɹ Ⱥ ɨɬ ɟɝɨ ɜɟɪɨɹɬɧɨɫɬɢ p A ɩɨ ɫɥɟɞɭɸɳɟɣ ɩɪɢɛɥɢɠɟɧɧɨɣ ɮɨɪɦɭɥɟ
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ɡɞɟɫɶ ) ±ɤɚɤ ɢ ɜɵɲɟ ± ɢɧɬɟɝɪɚɥɶɧɚɹ ɮɭɧɤɰɢɹ Ʌɚɩɥɚɫɚ |
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Ⱦɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɡɚɩɢɲɟɦ ɧɟɪɚɜɟɧɫɬɜɨ _ m |
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ɫ k n p H k |
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ɂɡ ɪɟɡɭɥɶɬɚɬɨɜ ɩ ɫɥɟɞɭɟɬ ɭɩɨɦɹɧɭɬɵɣ ɜɵɲɟ ɡɚɤɨɧ ɛɨɥɶɲɢɯ ɱɢ ɫɟɥ Ȼɟɪɧɭɥɥɢ
Теорема. ɉɭɫɬɶ w A mA ± ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɱɚɫɬɨɬɚ ɫɨɛɵɬɢɹ Ⱥ ɜ ɩ |
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ɱɚɫɬɶ ɫɬɪɟɦɢɬɫɹ ɤ ɡɧɚɱɟɧɢɸ Ɏ f |
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ȼ ɫɜɹɡɢ ɫ ɪɚɜɟɧɫɬɜɨɦ ɝɨɜɨɪɹɬ ɬɚɤɠɟ ɱɬɨ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɱɚɫɬɨɬɚ ɫɨ |
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ɛɵɬɢɹ Ⱥ ɫɯɨɞɢɬɫɹ ɩɨ ɜɟɪɨɹɬɧɨɫɬɢ ɩɪɢ ɩof ɤ ɜɟɪɨɹɬɧɨɫɬɢ ɪ $ |
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ɉɭɫɬɶ ɜ ɤɚɠɞɨɦ ɨɩɵɬɟ ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɛɵɬɢɹ A ɩɨɫɬɨɹɧɧɚ ɧɨ ɨɬ ɫɟɪɢɢ ɤ ɫɟɪɢɢ ɨɩɵɬɨɜ ɫ ɪɨɫɬɨɦ ɢɯ ɤɨɥɢɱɟɫɬɜɚ ɩ ɨɧɚ ɦɟɧɹɟɬɫɹ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨ ɧɚɥɶɧɨ ɤɨɥɢɱɟɫɬɜɭ ɨɩɵɬɨɜ ɞɪɭɝɢɦɢ ɫɥɨɜɚɦɢ ɡɧɚɱɟɧɢɟ ɩɪɨɢɡɜɟɞɟɧɢɹ O np
ɨɫɬɚɺɬɫɹ ɩɨɫɬɨɹɧɧɵɦ ɉɪɢ ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɵɯ ɭɫɥɨɜɢɹɯ ɢɦɟɟɬ ɦɟɫɬɨ ɫɨɨɬɧɨ ɲɟɧɢɟ
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ɉɪɢ ɜɵɱɢɫɥɟɧɢɢ ɩɪɟɞɟɥɚ ɩɪɢ n of ɡɚɦɟɬɢɦ ɱɬɨ ɨɬɧɨɲɟɧɢɟ Ok ɨɫɬɚɟɬɫɹ k
ɩɨɫɬɨɹɧɧɵɦ ɩɪɨɢɡɜɟɞɟɧɢɟ ɠɟ ɜ ɫɤɨɛɤɚɯ ɫɨɞɟɪɠɢɬ k ɦɧɨɠɢɬɟɥɟɣ ɤɚɠɞɵɣ ɢɡ ɤɨɬɨɪɵɯ ɫɬɪɟɦɢɬɫɹ ɤ ɟɞɢɧɢɰɟ ɉɨɫɥɟɞɧɢɣ ɦɧɨɠɢɬɟɥɶ ɫɨɞɟɪɠɢɬ ɜ ɫɤɨɛɤɚɯ
34
ɜɵɪɚɠɟɧɢɟ ɜɢɞɚ t 
t ɝɞɟ t On ɬɚɤ ɱɬɨ t o ɟɫɥɢ n of ɉɪɟɞɟɥ ɷɬɨɝɨ ɜɵɪɚɠɟɧɢɹ ɜɬɨɪɨɣ ɡɚɦɟɱɚɬɟɥɶɧɵɣ ɩɪɟɞɟɥ ɟɫɬɶ ɱɢɫɥɨ ɟ ɉɨɤɚɡɚɬɟɥɶ ɠɟ ɫɬɟɩɟɧɢ
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ɩɪɢ ɷɬɨɦ ɫɬɪɟɦɢɬɫɹ ɤ ɡɧɚɱɟɧɢɸ O Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɪɟɞɟɥɶɧɵɦ ɡɧɚɱɟɧɢɟɦ ɩɪɨɢɡɜɟɞɟɧɢɹ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ
ɹɜɥɹɟɬɫɹ Ok k e O ɱɟɦ ɢ ɞɨɤɚɡɚɧɨ ɭɬɜɟɪɠɞɟɧɢɟ k
ɍɫɬɚɧɨɜɥɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ ɨɡɧɚɱɚɟɬ ɱɬɨ ɟɫɥɢ ɜɟɪɨɹɬɧɨɫɬɶ ɪ ɩɨɹɜɥɟɧɢɹ ɫɨ ɛɵɬɢɹ Ⱥ ɜ ɤɚɠɞɨɦ ɢɡ n ɨɩɵɬɨɜ ɦɚɥɚ ɚ ɩɪɢ ɷɬɨɦ n ± ɜɟɥɢɤɨ ɬɨ ɢɦɟɟɬ ɦɟɫɬɨ ɩɪɢ ɛɥɢɠɟɧɧɚɹ ɮɨɪɦɭɥɚ
Oke O
Pn k | k
ɜ ɤɨɬɨɪɨɣ O np ȼɟɪɨɹɬɧɨɫɬɶ ɠɟ ɬɨɝɨ ɱɬɨ ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɦ ɱɢɫɥɟ ɨɩɵɬɨɜ ɫɨɛɵɬɢɟ
ɩɪɨɢɡɨɣɞɟɬ ɧɟ ɦɟɧɟɟ m ɢ ɧɟ ɛɨɥɟɟ m ɪɚɡ ɦɨɠɧɨ ɧɚɣɬɢ ɫɪ ɩ ɩɨ ɩɪɢ ɛɥɢɠɟɧɧɨɣ ɮɨɪɦɭɥɟ
m Ok
Pn m d m d m | k¦m k e O
12. Схема Бернулли: типовые задачи
Задача 12.1. ɇɚɣɬɢ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɫɨɛɵɬɢɟ Ⱥ ɩɨɹɜɢɬɫɹ ɜ ɩɹɬɢ ɧɟɡɚɜɢɫɢɦɵɯ ɨɩɵɬɚɯ
ɚ ɞɜɚ ɪɚɡɚ ɛ ɦɟɧɟɟ ɞɜɭɯ ɪɚɡ ɟɫɥɢ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɹɜɥɟɧɢɹ ɫɨɛɵɬɢɹ Ⱥ ɜ ɨɞɧɨɦ ɨɩɵɬɟ
ɪ
Ɋɟɲɟɧɢɟ
ɚ ɉɭɫɬɶ ɫɨɛɵɬɢɟ ȼ ɫɨɫɬɨɢɬ ɜ ɩɨɹɜɥɟɧɢɢ Ⱥ ɪɨɜɧɨ ɞɜɚ ɪɚɡɚ ɜ ɩɹɬɢ ɨɩɵɬɚɯ Ɍɨɝɞɚ ɩɨ ɮɨɪɦɭɥɟ Ȼɟɪɧɭɥɥɢ
P B P C
ɛ ȿɫɥɢ ɫɨɛɵɬɢɟ ɋ ɨɡɧɚɱɚɟɬ ɩɨɹɜɥɟɧɢɟ Ⱥ ɦɟɧɟɟ ɞɜɭɯ ɪɚɡ ɬɨ ɟɫɬɶ ɢɥɢ ɧɢ ɪɚ ɡɭ k ɢɥɢ ɨɞɢɧ ɪɚɡ k ɬɨ
P C P P C C
35
Задача 12.2. Ɏɭɬɛɨɥɶɧɚɹ ɤɨɦɚɧɞɚ ɢɝɪɚɟɬ ɫ ɪɚɜɧɨɫɢɥɶɧɵɦ ɫɨɩɟɪɧɢɤɨɦ ɫɟɪɢɸ ɦɚɬɱɟɣ ɜ ɫɥɭɱɚɟ ɧɢɱɶɟɣ ɧɚɡɧɚɱɚɸɬɫɹ ɛɭɥɥɢɬɵ ɑɬɨ ɜɟɪɨɹɬɧɟɟ ɜɵɢɝɪɚɬɶ ɞɜɚ ɦɚɬɱɚ ɢɡ ɱɟɬɵɪɟɯ ɢɥɢ ɬɪɢ ɢɡ ɲɟɫɬɢ"
Ɋɟɲɟɧɢɟ ɂɦɟɟɦ ɫɯɟɦɭ Ȼɟɪɧɭɥɥɢ ɜ ɤɨɬɨɪɨɣ ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɛɵɬɢɹ ɜ ɟɞɢ ɧɢɱɧɨɦ ɨɩɵɬɟ ɜɵɢɝɪɵɲɚ ɜ ɦɚɬɱɟ p q p ɋɪɚɜɧɢɦ P ɢ P
ɂɦɟɟɦ
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɜɟɪɨɹɬɧɨɫɬɶ ɜɵɢɝɪɚɬɶ ɦɚɬɱɚ ɢɡ ± ɜɵɲɟ
Задача 12.3. Ɍɟɫɬ ɩɨ ɦɚɬɟɦɚɬɢɤɟ ɫɞɚɸɬ ɫɬɭɞɟɧɬɨɜ ȼɟɪɨɹɬɧɨɫɬɶ ɩɪɟ ɨɞɨɥɟɬɶ ɩɨɪɨɝɨɜɨɟ ɡɧɚɱɟɧɢɟ ɞɥɹ ɤɚɠɞɨɝɨ ɢɡ ɧɢɯ ɇɚɣɬɢ ɧɚɢɜɟɪɨɹɬɧɟɣɲɟɟ ɱɢɫɥɨ ɭɫɩɟɲɧɨ ɩɪɨɲɟɞɲɢɯ ɬɟɫɬɢɪɨɜɚɧɢɟ ɢ ɟɝɨ ɜɟɪɨɹɬɧɨɫɬɶ
Ɋɟɲɟɧɢɟ Ɍɚɤ ɤɚɤ n p ɬɨ ɩɨ ɮɨɪɦɭɥɟ ɢɦɟɟɦ
d ɬɨ d ɬ ɟ ɬɨ
Ɍɟɩɟɪɶ
P C |
Задача 12.4.
ɚ ȼɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɧɚ ɫɬɪɚɧɢɰɟ ɪɭɤɨɩɢɫɢ ɢɦɟɟɬɫɹ ɯɨɬɹ ɛɵ ɨɞɧɚ ɨɩɟ ɱɚɬɤɚ ɪɚɜɧɚ ɇɚɣɬɢ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɢɡ ɫɬɪɚɧɢɰ ɤɧɢɝɢ ɪɨɜɧɨ ɫɨ ɞɟɪɠɚɬ ɨɩɟɱɚɬɤɢ
ɛ ɉɪɢ ɬɟɯ ɠɟ ɭɫɥɨɜɢɹɯ ɧɚɣɬɢ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɛɭɞɟɬ ɧɟ ɦɟɧɟɟ ɢ ɧɟ ɛɨɥɟɟ ɫɬɪɚɧɢɰ ɫ ɨɩɟɱɚɬɤɚɦɢ
Ɋɟɲɟɧɢɟ ɚ Ɇɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɥɨɤɚɥɶɧɭɸ ɮɨɪɦɭɥɭ Ʌɚɩɥɚɫɚ ɝɞɟ
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ɉɨɫɤɨɥɶɤɭ M ɬɨ ɛɭɞɟɦ ɢɦɟɬɶ |
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36
ɛ ɂɫɩɨɥɶɡɭɟɦ ɢɧɬɟɝɪɚɥɶɧɭɸ ɮɨɪɦɭɥɭ Ʌɚɩɥɚɫɚ ɜ ɤɨɬɨɪɨɣ
n ɪ q
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ɇɚɯɨɞɢɦ Ɏ ± Ɏ ± Ɏ ɋɥɟɞɨɜɚɬɟɥɶɧɨ
Ɋ d k d Ɏ ± Ɏ ± Ɏ Ɏ
Задача 12.5. ɋɤɨɥɶɤɨ ɪɚɡ ɧɚɞɨ ɛɪɨɫɢɬɶ ɦɨɧɟɬɭ ɱɬɨɛɵ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ ɪ ɭɬɜɟɪɠɞɚɬɶ ɱɬɨ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɱɚɫɬɨɬɚ ɜɵɩɚɞɟɧɢɹ ɝɟɪɛɚ ɨɬɤɥɨɧɢɥɚɫɶ ɨɬ ɧɟ ɛɨɥɟɟ ɱɟɦ ɧɚ "
Ɋɟɲɟɧɢɟ ɋɨɝɥɚɫɧɨ ɩɪɢɛɥɢɠɟɧɧɨɣ ɮɨɪɦɭɥɟ ɢɦɟɟɦ
P _ m |
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ɝɞɟ ɬ ± ɱɢɫɥɨ ɩɨɹɜɥɟɧɢɣ ɝɟɪɛɚ ɩɪɢ ɩ ɩɨɞɛɪɚɫɵɜɚɧɢɹɯ ɦɨɧɟɬɵ Ɉɬɫɸɞɚ| ) ɩ ɢɥɢ ) ɩ |
ɉɨ ɬɚɛɥɢɰɟ ɡɧɚɱɟɧɢɣ ɢɧɬɟɝɪɚɥɶɧɨɣ ɮɭɧɤɰɢɢ Ʌɚɩɥɚɫɚ ɧɚɯɨɞɢɦ ɡɧɚɱɟɧɢɟ ɚɪɝɭɦɟɧɬɚ ɢɧɬɟɝɪɚɥɶɧɨɣ ɮɭɧɤɰɢɢ Ʌɚɩɥɚɫɚ ɩ | ɨɬɤɭɞɚ ɩ | ɂɬɚɤ ɦɨ ɧɟɬɭ ɧɚɞɨ ɛɪɨɫɢɬɶ ɪɚɡ
Задача 12.6. ȼɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɚɛɨɧɟɧɬ ɧɟɩɪɚɜɢɥɶɧɨ ɧɚɛɟɪɟɬ ɬɟɥɟ ɮɨɧɧɵɣ ɧɨɦɟɪ ɩɪɢɧɢɦɚɟɬɫɹ ɞɥɹ ɜɫɟɯ ɚɛɨɧɟɧɬɨɜ ɪɚɜɧɨɣ Ɉɩɪɟɞɟɥɢɬɶ ɜɟ ɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɫɪɟɞɢ ɩɪɨɢɡɜɟɞɟɧɧɵɯ ɧɟɡɚɜɢɫɢɦɨ ɨɞɢɧ ɨɬ ɞɪɭɝɨɝɨ ɜɵ ɡɨɜɨɜ ɪɨɜɧɨ ɨɞɢɧ ɚɛɨɧɟɧɬ ɧɚɛɟɪɟɬ ɧɟɩɪɚɜɢɥɶɧɨ ɬɟɥɟɮɨɧɧɵɣ ɧɨɦɟɪ
Ɋɟɲɟɧɢɟ ɑɢɫɥɨ ɨɩɵɬɨɜ n ɜɟɥɢɤɨ ɜɟɪɨɹɬɧɨɫɬɶ ɧɟɜɟɪɧɨɝɨ ɧɚɛɨɪɚ ɧɨ ɦɟɪɚ ɜ ɤɚɠɞɨɦ ɨɩɵɬɟ ɪ ± ɦɚɥɚ ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɩɪɢɦɟɧɹɟɦ ɮɨɪɦɭɥɭ
ɉɭɚɫɫɨɧɚ ɜ ɤɨɬɨɪɨɣ O np |
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Задача 12.7. ȼ ɬɟɱɟɧɢɟ ɝɨɞɚ ɢɡ ɚɷɪɨɩɨɪɬɚ ɝɨɪɨɞɚ 1 ɨɬɩɪɚɜɥɹɟɬɫɹɚɜɢɚɪɟɣɫɨɜ ȼɟɪɨɹɬɧɨɫɬɶ ɡɚɞɟɪɠɤɢ ɤɚɠɞɨɝɨ ɜɵɥɟɬɚ ɩɨ ɦɟɬɟɨɭɫɥɨɜɢɹɦ ɪɚɜ ɧɚ Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɡɚɞɟɪɠɤɢ ɩɨ ɦɟɬɟɨɭɫɥɨɜɢɹɦ ɜ ɬɟɱɟɧɢɟ ɝɨɞɚ ɧɟ ɦɟ ɧɟɟ ɪɟɣɫɨɜ"
Ɋɟɲɟɧɢɟ ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ ɜ ɟɞɢɧɢɱ ɧɨɦ ɨɩɵɬɟ ɡɚɞɟɪɠɤɢ ɪɟɣɫɚ ɩɨ ɦɟɬɟɨɭɫɥɨɜɢɹɦ ɦɚɥɚ ɪ ɬɨɝɞɚ ɤɚɤ ɱɢɫɥɨ ɨɩɵɬɨɜ ɱɢɫɥɨ ɪɟɣɫɨɜ ɜɟɥɢɤɨ n ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɜ ɪɚɫɱɟɬɚɯ ɜɨɡɦɨɠɧɨ
37
ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɮɨɪɦɭɥɵ ɉɭɚɫɫɨɧɚ ɋɨɛɵɬɢɟ ɡɚɞɟɪɠɤɢ ɧɟ ɦɟɧɟɟ ɪɟɣɫɨɜ ɩɪɨɬɢ
ɜɨɩɨɥɨɠɧɨ ɫɨɛɵɬɢɸ ɡɚɞɟɪɠɤɢ k d ɪɟɣɫɨɜ ɂɦɟɟɦ O |
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КОНТРОЛЬНЫЙ МОДУЛЬ
СЛУЧАЙНЫЕ СОБЫТИЯ
ɌȿɈɊȿɌɂɑȿɋɄɂȿ ɍɉɊȺɀɇȿɇɂə
ɂɫɩɨɥɶɡɭɹ ɬɚɛɥɢɰɵ ɢɫɬɢɧɧɨɫɬɢ ɞɨɤɚɡɚɬɶ ɪɚɜɟɧɫɬɜɚ
Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ A A
ȿɫɥɢ ɤɥɚɫɫɢɱɟɫɤɚɹ ɜɟɪɨɹɬɧɨɫɬɶ ɧɟɤɨɬɨɪɨɝɨ ɫɨɛɵɬɢɹ ɪɚɜɧɚ ɬɨ ɜɟɪɧɨ ɥɢ ɱɬɨ ɷɬɨ ɫɨɛɵɬɢɟ ± ɞɨɫɬɨɜɟɪɧɨɟ" ɉɨɱɟɦɭ"
Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɞɥɹ ɥɸɛɵɯ ɞɜɭɯ ɫɨɛɵɬɢɣ Ⱥ ɢ ȼ ɢɦɟɟɬ ɦɟɫɬɨ ɧɟɪɚɜɟɧɫɬɜɨ
P A B d P A Ɋ %
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Ɇɨɠɧɨ ɥɢ ɜ ɫɥɭɱɚɟ ɞɜɭɯ ɝɢɩɨɬɟɡ ɮɨɪɦɭɥɭ ɩɨɥɧɨɣ ɜɟɪɨɹɬɧɨɫɬɢ ɡɚɩɢɫɚɬɶ |
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Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɜ ɫɯɟɦɟ Ȼɟɪɧɭɥɥɢ ɜɟɪɨɹɬɧɨɫɬɶ U ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ Ⱥ ɯɨɬɹ ɛɵ ɨɞɢɧ ɪɚɡ ɦɨɠɟɬ ɛɵɬɶ ɜɵɱɢɫɥɟɧɚ ɜ ɜɢɞɟ U qn Ɂɞɟɫɶ ɪ ± ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɹɜɥɟɧɢɹ ɫɨɛɵɬɢɹ Ⱥ ɜ ɤɚɠɞɨɦ ɢɡ n ɨɩɵɬɨɜ q ɪ
ȼɟɪɨɹɬɧɨɫɬɶ ɪ ɩɨɹɜɥɟɧɢɹ ɫɨɛɵɬɢɹ Ⱥ ɜ ɤɚɠɞɨɦ ɨɩɵɬɟ ɩɨɫɬɨɹɧɧɚ Ɉɩɵɬɵ ɩɪɨɜɨɞɹɬ ɞɨ ɩɟɪɜɨɝɨ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ ɑɟɦɭ ɪɚɜɧɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɛɭɞɟɬ ɩɪɨɜɟɞɟɧɨ ɪɨɜɧɨ ɩ ɨɩɵɬɨɜ
Ⱦɨɤɚɡɚɬɶ ɱɬɨ ɟɫɥɢ ɜɟɪɨɹɬɧɨɫɬɶ ɪ ɩɨɹɜɥɟɧɢɹ ɫɨɛɵɬɢɹ Ⱥ ɜ ɤɚɠɞɨɦ ɢɡ n ɨɩɵɬɨɜ ɦɚɥɚ Q – ɜɟɥɢɤɨ ɢ O np ɬɨ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɹɜɥɟɧɢɹ ɫɨɛɵɬɢɹ Ⱥ ɯɨɬɹ ɛɵ ɨɞɢɧ ɪɚɡ ɦɨɠɟɬ ɛɵɬɶ ɧɚɣɞɟɧɚ ɩɨ ɩɪɢɛɥɢɠɟɧɧɨɣ ɮɨɪɦɭɥɟ
P k t | e O
38
ɁȺȾȺɑɂ ȾɅə ɋȺɆɈɋɌɈəɌȿɅɖɇɈȽɈ Ɋȿɒȿɇɂə
Ɉɛɪɚɡɭɸɬ ɥɢ ɩɨɥɧɭɸ ɝɪɭɩɩɭ ɫɥɟɞɭɸɳɢɟ ɫɨɛɵɬɢɹ Ⱥ ± ɞɜɚ ɩɨɩɚɞɚɧɢɹ ɜ ɦɢɲɟɧɶ ɩɪɢ ɞɜɭɯ ɜɵɫɬɪɟɥɚɯ ȼ ± ɧɢ ɨɞɧɨɝɨ ɩɨɩɚɞɚɧɢɹ ɩɪɢ ɬɟɯ ɠɟ ɞɜɭɯ ɜɵɫɬɪɟ ɥɚɯ"
Ɋɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɫɨɛɵɬɢɹ Ⱥ ± ɩɟɪɜɨɟ ɢɡ ɩɨɥɭɱɟɧɧɵɯ ɷɥɟɤ ɬɪɨɧɧɵɯ ɩɢɫɟɦ ɫɨɞɟɪɠɢɬ ɧɚɜɹɡɱɢɜɭɸ ɪɟɤɥɚɦɭ ɋɉȺɆ ȼ ± ɜɬɨɪɨɟ ɩɢɫɶɦɨ ɫɨ ɞɟɪɠɢɬ ɋɉȺɆ ȼɵɪɚɡɢɬɶ ɫ ɩɨɦɨɳɶɸ ɨɩɟɪɚɰɢɣ ɫɥɨɠɟɧɢɹ ɢ ɭɦɧɨɠɟɧɢɹ ɱɟɪɟɡ ɫɨɛɵɬɢɹ Ⱥ ɢ ȼ ɢ ɢɥɢ ɢɦ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɫɥɟɞɭɸɳɢɟ ɫɨɛɵɬɢɹ
ɚ ɫɨɛɵɬɢɟ ɋ ± ɧɢ ɨɞɧɨ ɢɡ ɩɢɫɟɦ ɧɟ ɫɨɞɟɪɠɢɬ ɋɉȺɆ ɛ ɯɨɬɹ ɛɵ ɨɞɧɨ ɩɢɫɶɦɨ ɫɨɞɟɪɠɢɬ ɋɉȺɆ ɜ ɬɨɥɶɤɨ ɨɞɧɨ ɩɢɫɶɦɨ ɫɨɞɟɪɠɢɬ ɋɉȺɆȼɟɪɨɹɬɧɨɫɬɶ ɞɨɠɞɥɢɜɨɣ ɩɨɝɨɞɵ ɜ ɩɪɟɞɫɬɨɹɳɢɣ ɜɵɯɨɞɧɨɣ ɞɟɧɶ ɪɚɜ
ɧɚ ȼɟɪɨɹɬɧɨɫɬɶ ɭɞɚɱɧɨɣ ɪɵɛɚɥɤɢ ɜ ɞɨɠɞɥɢɜɭɸ ɩɨɝɨɞɭ ɪɚɜɧɚ ɚ ɜ ɹɫɧɭɸ ɩɨɝɨɞɭ ± Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɱɬɨ ɜ ɩɪɟɞɫɬɨɹɳɢɣ ɜɵɯɨɞɧɨɣ ɪɵɛɚɥɤɚ ɛɭɞɟɬ ɭɞɚɱɧɨɣ
Ɂɚɹɜɤɢ ɪɚɛɨɬɨɞɚɬɟɥɟɣ ɧɚ ɫɩɟɰɢɚɥɢɫɬɨɜ ɢɧɠɟɧɟɪɧɵɯ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɢ ɸɪɢɞɢɱɟɫɤɢɟ ɧɚɩɪɚɜɥɟɧɢɣ ɩɨɫɬɭɩɚɸɬ ɧɚ ɛɢɪɠɭ ɜ ɨɬɧɨɲɟɧɢɢ ȼɟ ɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɩɪɟɬɟɧɞɟɧɬ ɧɚ ɜɚɤɚɧɫɢɸ ɢɧɠɟɧɟɪɚ ɭɞɨɜɥɟɬɜɨɪɢɬ ɬɪɟɛɨɜɚ ɧɢɹɦ ɪɚɛɨɬɨɞɚɬɟɥɹ ɪɚɜɧɚ ɧɚ ɜɚɤɚɧɫɢɸ ɷɤɨɧɨɦɢɫɬɚ ± ɧɚ ɜɚɤɚɧɫɢɸ ɸɪɢ ɫɬɚ ±
ɇɚɣɬɢ ɜɟɪɨɹɬɧɨɫɬɶ ɱɬɨ ɚ ɫɥɭɱɚɣɧɨ ɜɵɛɪɚɧɧɵɣ ɧɚ ɛɢɪɠɟ ɩɪɟɬɟɧɞɟɧɬ ɭɫɬɪɨɢɬɫɹ ɧɚ ɪɚɛɨɬɭ ɩɨ ɫɜɨɟɣ
ɫɩɟɰɢɚɥɶɧɨɫɬɢ ɛ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɭɫɬɪɨɢɜɲɢɣɫɹ ɧɚ ɪɚɛɨɬɭ ɫɩɟɰɢɚɥɢɫɬ ± ɷɤɨɧɨɦɢɫɬɂɦɟɟɬɫɹ ɞɜɚɞɰɚɬɢɞɨɥɥɚɪɨɜɵɯ ɤɭɩɸɪ ɢɡ ɤɨɬɨɪɵɯ ɤɭɩɸɪɵ ɮɚɥɶɲɢ
ɜɵɟ ɇɚɭɝɚɞ ɩɨɨɱɟɪɟɞɧɨ ɢɡɜɥɟɤɚɸɬ ɞɜɟ ɤɭɩɸɪɵ ɢ ɨɬɵɫɤɢɜɚɸɬ ɜɟɪɨɹɬɧɨɫɬɶ ɫɨ ɛɵɬɢɹ Ⱥ ɫɨɫɬɨɹɳɟɝɨ ɜ ɬɨɦ ɱɬɨ ɨɛɟ ɷɬɢ ɤɭɩɸɪɵ ɨɤɚɠɭɬɫɹ ɮɚɥɶɲɢɜɵɦɢ Ɇɨɠɧɨ ɥɢ ɩɪɢɦɟɧɹɬɶ ɮɨɪɦɭɥɭ Ȼɟɪɧɭɥɥɢ ɟɫɥɢ ɚ ɤɭɩɸɪɚ ɩɨɫɥɟ ɢɡɜɥɟɱɟɧɢɹ ɢ ɩɪɨɜɟɪɤɢ ɜɨɡɜɪɚɳɚɟɬɫɹ ɜ ɩɚɱɤɭ ɛ ɜɵɛɨɪɤɚ ɛɟɡɜɨɡɜɪɚɬɧɚɹ
ɇɚɣɬɢ Ɋ Ⱥ ɜ ɤɚɠɞɨɦ ɢɡ ɫɥɭɱɚɟɜ ɚ ɢ ɛȼɟɪɨɹɬɧɨɫɬɶ ɩɪɨɞɚɬɶ ɩɨ ɨɩɬɢɦɚɥɶɧɨɣ ɰɟɧɟ ɤɚɠɞɵɣ ɢɡ ɩɹɬɢ ɩɚɤɟɬɨɜ ɚɤ
ɰɢɣ ɜ ɩɟɪɢɨɞ ɢɯ ɩɚɞɟɧɢɹ ɪɚɜɧɚ Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɩɪɨɞɚɠɢ ɩɨ ɨɩɬɢɦɚɥɶ ɧɨɣ ɰɟɧɟ ɛɨɥɶɲɟɣ ɱɚɫɬɢ ɩɚɤɟɬɚ"
ȼ ɩɪɹɦɨɭɝɨɥɶɧɢɤ ɜɩɢɫɚɧɵ ɞɜɟ ɨɤɪɭɠɧɨɫɬɢ ɪɚɜɧɨɝɨ ɪɚɞɢɭɫɚ ɤɚɫɚɸɳɢɟɫɹ ɞɪɭɝ ɞɪɭɝɚ ɜɧɟɲɧɢɦ ɨɛɪɚɡɨɦ ȼ ɩɪɹɦɨɭɝɨɥɶɧɢɤ ɫɥɭɱɚɣɧɵɦ ɨɛɪɚɡɨɦ ɛɪɨɲɟɧɚ ɬɨɱɤɚ Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɱɬɨ ɨɧɚ ɧɟ ɩɨɩɚɞɟɬ ɧɢ ɜ ɨɞɢɧ ɢɡ ɤɪɭɝɨɜ"
ȼ ɫɟɦɶɟ ɞɟɬɟɣ ɜɟɪɨɹɬɧɨɫɬɶ ɪɨɠɞɟɧɢɹ ɦɚɥɶɱɢɤɚ ɜ ɞɚɧɧɨɣ ɦɟɫɬɧɨɫɬɢ ɪɚɜɧɚ ɇɚɣɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɫɥɟɞɭɸɳɢɯ ɫɨɛɵɬɢɣ
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ɚ ɜ ɫɟɦɶɟ ɞɜɟ ɞɟɜɨɱɤɢ ɛ ɜ ɫɟɦɶɟ ɧɟ ɦɟɧɟɟ ɞɜɭɯ ɞɟɜɨɱɟɤ ɜ ɜ ɫɟɦɶɟ ɦɚɥɶɱɢɤɨɜ ɛɨɥɶɲɟ ɱɟɦ ɞɟɜɨɱɟɤ
ȼ ɞɚɧɧɨɣ ɦɟɫɬɧɨɫɬɢ ɥɟɜɲɢ ɫɨɫɬɚɜɥɹɸɬ ɧɚɫɟɥɟɧɢɹ Ʉɚɤɨɜɚ ɜɟɪɨɹɬ ɧɨɫɬɶ ɱɬɨ ɧɚ ɮɚɤɭɥɶɬɟɬɟ ɝɞɟ ɨɛɭɱɚɸɬɫɹ ɱɟɥɨɜɟɤ ɨɤɚɠɭɬɫɹ ɧɟ ɦɟɧɟɟ ɥɟɜ ɲɟɣ"
ɋɬɭɞɟɧɬ ɨɞɢɧɚɤɨɜɨ ɩɥɨɯɨ ɩɨɞɝɨɬɨɜɢɥɫɹ ɤ ɤɚɠɞɨɦɭ ɢɡ ɬɪɟɯ ɷɤɡɚɦɟɧɨɜ ɋ ɤɚɤɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ ɨɧ ɫɞɚɟɬ ɤɚɠɞɵɣ ɷɤɡɚɦɟɧ ɟɫɥɢ ɯɨɬɹ ɛɵ ɨɞɢɧ ɢɡ ɧɢɯ ɨɧ ɫɞɚɫɬ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ
Ʉɚɠɞɵɣ ɢɡ ɬɪɟɯ ɧɟɡɚɜɢɫɢɦɨ ɪɚɛɨɬɚɸɳɢɯ ɫɢɝɧɚɥɢɡɚɬɨɪɨɜ ɫɜɨɟɜɪɟɦɟɧɧɨ ɫɨɨɛɳɚɟɬ ɨ ɧɚɪɭɲɟɧɢɢ ɡɚɞɚɧɧɨɝɨ ɪɟɠɢɦɚ ɪɚɛɨɬɵ ɪɟɚɤɬɨɪɚ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ ɫɨ ɨɬɜɟɬɫɬɜɟɧɧɨ p p p Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɩɪɢ ɧɚɪɭɲɟɧɢɢ ɡɚɞɚɧɧɨɝɨ ɪɟɠɢɦɚ ɪɚɛɨɬɵ ɫɢɝɧɚɥɚ ɧɟ ɩɨɫɬɭɩɢɬ"
ɋɪɟɞɢ ɩɹɬɢ ɨɞɢɧɚɤɨɜɵɯ ɩɨ ɜɧɟɲɧɟɦɭ ɜɢɞɭ ɫɚɠɟɧɰɟɜ ɬɪɢ ± ɷɥɢɬɧɵɯ ɇɚɭɝɚɞ ɜɡɹɬɵ ɞɜɚ ɫɚɠɟɧɰɚ Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɱɬɨ ɪɨɜɧɨ ɨɞɢɧ ɢɡ ɧɢɯ ɷɥɢɬ ɧɵɣ"
ȼɟɪɨɹɬɧɨɫɬɢ ɫɜɨɟɜɪɟɦɟɧɧɨɝɨ ɩɨɫɬɭɩɥɟɧɢɹ ɩɥɚɬɟɠɚ ɧɚ ɤɚɪɬɵ ɩɪɢɨɛɪɟ ɬɺɧɧɵɟ ɜ ɛɚɧɤɚɯ Ⱥ Ȼ ȼ ɪɚɜɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ p p p ɇɚ ɤɚɠɞɭɸ ɢɡ ɤɚɪɬ ɛɚɧɤɨɜ Ⱥ Ȼ ȼ ɩɟɪɟɱɢɫɥɟɧɵ ɫɪɟɞɫɬɜɚ Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɪɨɜɧɨ ɧɚ ɞɜɟ ɤɚɪɬɵ ɩɟɪɟɜɨɞ ɩɨɫɬɭɩɢɬ ɫɜɨɟɜɪɟɦɟɧɧɨ"
ȼɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɤɚɠɞɵɣ ɞɚɧɧɵɣ ɤɥɢɟɧɬ ɛɚɧɤɚ ɩɨɬɪɟɛɭɟɬ ɜ ɬɟɱɟ ɧɢɟ ɤɚɥɟɧɞɚɪɧɨɝɨ ɝɨɞɚ ɡɚɤɪɵɬɢɹ ɫɜɨɟɝɨ ɥɢɰɟɜɨɝɨ ɫɱɟɬɚ ɪɚɜɧɚ Ʉɚɤɨɜɚ ɜɟ ɪɨɹɬɧɨɫɬɶ ɱɬɨ ɫɪɟɞɢ ɤɥɢɟɧɬɨɜ ɨɬɞɟɥɟɧɢɹ ɫɛɟɪɛɚɧɤɚ ɬɪɨɟ ɩɨɬɪɟɛɭɸɬ ɜ ɬɟ ɱɟɧɢɟ ɝɨɞɚ ɡɚɤɪɵɬɢɹ ɥɢɰɟɜɵɯ ɫɱɟɬɨɜ" Ɉɬɜɟɬ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ ɞɟɫɹɬɢɱɧɨɣ ɞɪɨɛɢ ɩɪɢɛɥɢɠɟɧɧɨ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ
ɋɤɨɥɶɤɨ ɫɥɟɞɭɟɬ ɪɚɡɦɟɫɬɢɬɶ ɥɸɦɢɧɟɫɰɟɧɬɧɵɯ ɥɚɦɩ ɧɚ ɩɨɬɨɥɤɟ ɨɮɢɫɚ ɱɬɨɛɵ ɨɮɢɫ ɛɵɥ ɨɫɜɟɳɟɧ ɯɨɬɹ ɛɵ ɨɞɧɨɣ ɥɚɦɩɨɣ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ ɟɫɥɢ ɜɟɪɨɹɬɧɨɫɬɶ ɩɟɪɟɝɨɪɚɧɢɹ ɤɚɠɞɨɣ ɥɚɦɩɵ ɪɚɜɧɚ "
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